Jump Game V - Minimum Jumps to Reach End (Java, Time Complexity: O(n))

## Understanding the Problem The core challenge of this problem is to determine the minimum number of jumps required to reach the last index of the array. Each element in the array specifies the maximum number of steps you can jump forward from that position. The problem is significant in scenarios like game development, where you need to determine the shortest path or minimum moves to reach a goal. ### Potential Pitfalls and Misconceptions - Misunderstanding the jump length: Each element represents the maximum jump length, not the exact number of steps you must take. - Overlooking edge cases: Arrays with a single element or elements that make it impossible to reach the end. ## Approach ### Naive Solution A naive approach would involve exploring all possible jumps from each index and keeping track of the minimum jumps required to reach the end. This approach, however, is not optimal due to its high time complexity. ### Optimized Solution A more efficient approach involves using a greedy algorithm. The idea is to keep track of the farthest point that can be reached and the end of the current jump range. When the end of the current jump range is reached, increment the jump count and update the range. ### Thought Process 1. Initialize variables to keep track of the current end of the jump range, the farthest point that can be reached, and the number of jumps. 2. Iterate through the array, updating the farthest point that can be reached. 3. When the current index reaches the end of the current jump range, increment the jump count and update the jump range to the farthest point. ## Algorithm 1. Initialize `jumps` to 0, `currentEnd` to 0, and `farthest` to 0. 2. Iterate through the array up to the second last element: - Update `farthest` to the maximum of `farthest` and `i + nums[i]`. - If the current index `i` reaches `currentEnd`, increment `jumps` and update `currentEnd` to `farthest`. 3. Return `jumps`. ## Code Implementation

public class JumpGameV {
    public int jump(int[] nums) {
        if (nums == null || nums.length == 0) return 0;
        
        int jumps = 0;
        int currentEnd = 0;
        int farthest = 0;
        
        for (int i = 0; i < nums.length - 1; i++) {
            // Update the farthest point that can be reached
            farthest = Math.max(farthest, i + nums[i]);
            
            // If we have reached the end of the current jump range
            if (i == currentEnd) {
                jumps++;
                currentEnd = farthest;
            }
        }
        
        return jumps;
    }
}

### Key Parts of the Code - `farthest = Math.max(farthest, i + nums[i]);`: This line updates the farthest point that can be reached from the current index. - `if (i == currentEnd)`: This condition checks if the current index has reached the end of the current jump range, prompting an increment in the jump count. ## Complexity Analysis - **Time Complexity**: O(n), where n is the length of the array. The array is traversed once. - **Space Complexity**: O(1), as no additional space proportional to the input size is used. ## Edge Cases - **Single Element Array**: The output should be 0 as no jumps are needed. - **Unreachable End**: Arrays where the end cannot be reached should be handled gracefully. ### Example Edge Cases - Input: `[0]`, Output: `0` - Input: `[1, 0, 1]`, Output: `1` ## Testing To test the solution comprehensively: - Use a variety of test cases, including simple, complex, and edge cases. - Example test cases: - `[2, 3, 1, 1, 4]` should return `2`. - `[1, 1, 1, 1]` should return `3`. - `[0]` should return `0`. ## Thinking and Problem-Solving Tips - Break down the problem into smaller parts and understand the role of each element in the array. - Practice similar problems to improve problem-solving skills and familiarity with greedy algorithms. ## Conclusion Understanding and solving the Jump Game V problem helps in developing skills in greedy algorithms and dynamic programming. Practice and exploration of similar problems can further enhance problem-solving abilities. ## Additional Resources - [LeetCode Problems](https://leetcode.com/problemset/all/) - [GeeksforGeeks Greedy Algorithms](https://www.geeksforgeeks.org/greedy-algorithms/) - [Introduction to Algorithms by Cormen et al.](https://mitpress.mit.edu/books/introduction-algorithms) By mastering this problem, you can improve your algorithmic thinking and be better prepared for coding interviews and competitive programming. Happy coding!